I am not clear on your planned schedule, but I suggest watching the snow pack reports ahead of your departure. It may take more than a few days of hiking for things to open up. Tioga's average opening date is May 27th and the median is May 24th, but mid to late June is common and it has been as late as July 1st. Tioga and Glacier Point Roads Opening & Closing Dates - Yosemite National Park (U.S. National Park Service)

In addition to that, the services in Yosemite other than in the Valley can open much later than the pass. The year we were there Tioga opened June 5th and services outside the Valley opened in July.

I have not looked into the likely detours for a Tioga closure, but it seems like you might miss some of the best scenery of the trip.

As far as a bit of hiking and/or sight seeing, even though I am usually not one to take days off from riding, we spent 5 days in the Yosemite Valley and found it well worth the time spent there. You can camp one night in the backpackers staging camp and then get to the climbers camp (Camp 4) early to be in line for a first come first served site. I forget what the limit was for a stay in the Valley, but it was in the 5-7 day range.